Optimal. Leaf size=95 \[ \frac {i \tanh ^{-1}\left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {3+2 \tan (c+d x)}}\right )}{\sqrt {2-3 i} d}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {3+2 \tan (c+d x)}}\right )}{\sqrt {2+3 i} d} \]
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Rubi [A]
time = 0.07, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3656, 924, 95,
214} \begin {gather*} \frac {i \tanh ^{-1}\left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {2 \tan (c+d x)+3}}\right )}{\sqrt {2-3 i} d}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {2 \tan (c+d x)+3}}\right )}{\sqrt {2+3 i} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 214
Rule 924
Rule 3656
Rubi steps
\begin {align*} \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {3+2 \tan (c+d x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {3+2 x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{2 (i-x) \sqrt {x} \sqrt {3+2 x}}+\frac {1}{2 \sqrt {x} (i+x) \sqrt {3+2 x}}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{(i-x) \sqrt {x} \sqrt {3+2 x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} (i+x) \sqrt {3+2 x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{i-(3+2 i) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {3+2 \tan (c+d x)}}\right )}{d}+\frac {\text {Subst}\left (\int \frac {1}{i+(3-2 i) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {3+2 \tan (c+d x)}}\right )}{d}\\ &=\frac {i \tanh ^{-1}\left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {3+2 \tan (c+d x)}}\right )}{\sqrt {2-3 i} d}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {3+2 \tan (c+d x)}}\right )}{\sqrt {2+3 i} d}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 95, normalized size = 1.00 \begin {gather*} \frac {i \text {ArcTan}\left (\frac {\sqrt {-2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {3+2 \tan (c+d x)}}\right )}{\sqrt {-2+3 i} d}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {3+2 \tan (c+d x)}}\right )}{\sqrt {2+3 i} d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(478\) vs.
\(2(77)=154\).
time = 0.87, size = 479, normalized size = 5.04
method | result | size |
derivativedivides | \(-\frac {3 \sqrt {\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right ) \left (\sqrt {13}\, \sqrt {2 \sqrt {13}+4}\, \arctan \left (\frac {\sqrt {\frac {\left (17 \sqrt {13}-52\right ) \tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right ) \left (52+17 \sqrt {13}\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}\, \sqrt {-4+2 \sqrt {13}}\, \left (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2-3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )}{56862 \tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}\right ) \sqrt {-4+2 \sqrt {13}}-2 \sqrt {2 \sqrt {13}+4}\, \arctan \left (\frac {\sqrt {\frac {\left (17 \sqrt {13}-52\right ) \tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right ) \left (52+17 \sqrt {13}\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}\, \sqrt {-4+2 \sqrt {13}}\, \left (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2-3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )}{56862 \tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}\right ) \sqrt {-4+2 \sqrt {13}}-8 \arctanh \left (\frac {6 \sqrt {13}\, \sqrt {\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right ) \sqrt {13}+34 \arctanh \left (\frac {6 \sqrt {13}\, \sqrt {\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \sqrt {3+2 \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+4}\, \left (17 \sqrt {13}-52\right )}\) | \(479\) |
default | \(-\frac {3 \sqrt {\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right ) \left (\sqrt {13}\, \sqrt {2 \sqrt {13}+4}\, \arctan \left (\frac {\sqrt {\frac {\left (17 \sqrt {13}-52\right ) \tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right ) \left (52+17 \sqrt {13}\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}\, \sqrt {-4+2 \sqrt {13}}\, \left (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2-3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )}{56862 \tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}\right ) \sqrt {-4+2 \sqrt {13}}-2 \sqrt {2 \sqrt {13}+4}\, \arctan \left (\frac {\sqrt {\frac {\left (17 \sqrt {13}-52\right ) \tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right ) \left (52+17 \sqrt {13}\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}\, \sqrt {-4+2 \sqrt {13}}\, \left (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2-3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )}{56862 \tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}\right ) \sqrt {-4+2 \sqrt {13}}-8 \arctanh \left (\frac {6 \sqrt {13}\, \sqrt {\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right ) \sqrt {13}+34 \arctanh \left (\frac {6 \sqrt {13}\, \sqrt {\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \sqrt {3+2 \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+4}\, \left (17 \sqrt {13}-52\right )}\) | \(479\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\tan {\left (c + d x \right )}}}{\sqrt {2 \tan {\left (c + d x \right )} + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 498 vs.
\(2 (67) = 134\).
time = 1.36, size = 498, normalized size = 5.24 \begin {gather*} -\frac {1}{676} \, \sqrt {2} {\left (\frac {3 \, {\left (2 \, \sqrt {325 \, \sqrt {13} - 1118} \arctan \left (\frac {13 \, \left (\frac {4}{13}\right )^{\frac {3}{4}} {\left (\left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + 1\right )}}{4 \, \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}}}\right ) + 2 \, \sqrt {325 \, \sqrt {13} - 1118} \arctan \left (-\frac {13 \, \left (\frac {4}{13}\right )^{\frac {3}{4}} {\left (\left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} - 1\right )}}{4 \, \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}}}\right ) - \sqrt {325 \, \sqrt {13} - 1118} \log \left (2 \, \left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + 2 \, \sqrt {\frac {1}{13}} + 1\right ) + \sqrt {325 \, \sqrt {13} - 1118} \log \left (-2 \, \left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + 2 \, \sqrt {\frac {1}{13}} + 1\right )\right )}}{d} - \frac {2 \, {\left (3 \, d^{2} \sqrt {325 \, \sqrt {13} - 1118} - 2 \, d \sqrt {325 \, \sqrt {13} - 1118} {\left | d \right |}\right )} \arctan \left (\frac {13 \, \left (\frac {4}{13}\right )^{\frac {3}{4}} {\left (\left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + \sqrt {-\frac {3}{2 \, \tan \left (d x + c\right ) + 3} + 1}\right )}}{4 \, \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}}}\right )}{d^{3}} - \frac {2 \, {\left (3 \, d^{2} \sqrt {325 \, \sqrt {13} - 1118} - 2 \, d \sqrt {325 \, \sqrt {13} - 1118} {\left | d \right |}\right )} \arctan \left (-\frac {13 \, \left (\frac {4}{13}\right )^{\frac {3}{4}} {\left (\left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} - \sqrt {-\frac {3}{2 \, \tan \left (d x + c\right ) + 3} + 1}\right )}}{4 \, \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}}}\right )}{d^{3}} + \frac {{\left (3 \, d^{2} \sqrt {325 \, \sqrt {13} - 1118} + 2 \, d \sqrt {325 \, \sqrt {13} - 1118} {\left | d \right |}\right )} \log \left (2 \, \left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} \sqrt {-\frac {3}{2 \, \tan \left (d x + c\right ) + 3} + 1} + 2 \, \sqrt {\frac {1}{13}} - \frac {3}{2 \, \tan \left (d x + c\right ) + 3} + 1\right )}{d^{3}} - \frac {{\left (3 \, d^{2} \sqrt {325 \, \sqrt {13} - 1118} + 2 \, d \sqrt {325 \, \sqrt {13} - 1118} {\left | d \right |}\right )} \log \left (-2 \, \left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} \sqrt {-\frac {3}{2 \, \tan \left (d x + c\right ) + 3} + 1} + 2 \, \sqrt {\frac {1}{13}} - \frac {3}{2 \, \tan \left (d x + c\right ) + 3} + 1\right )}{d^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.61, size = 207, normalized size = 2.18 \begin {gather*} \mathrm {atan}\left (\frac {\sqrt {3}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,\left (4-6{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,\sqrt {2\,\mathrm {tan}\left (c+d\,x\right )+3}\,\left (-4+6{}\mathrm {i}\right )}{2\,\mathrm {tan}\left (c+d\,x\right )-\sqrt {3}\,\sqrt {2\,\mathrm {tan}\left (c+d\,x\right )+3}+3}\right )\,\sqrt {\frac {-\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {\sqrt {3}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,\left (4+6{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,\sqrt {2\,\mathrm {tan}\left (c+d\,x\right )+3}\,\left (-4-6{}\mathrm {i}\right )}{2\,\mathrm {tan}\left (c+d\,x\right )-\sqrt {3}\,\sqrt {2\,\mathrm {tan}\left (c+d\,x\right )+3}+3}\right )\,\sqrt {\frac {-\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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